Learn Spanish (Latin America): Te aconsejo dirigir la orquesta hoy. - I advise you to conduct the orchestra today.

With verbs of influence like aconsejar (to advise), permitir (to allow), and mandar (to order), you actually have two structural choices. You can use the indirect object plus the infinitive (as seen here: Te aconsejo dirigir) or the indirect object plus que and the subjunctive (Te aconsejo que dirijas). Both are perfectly correct and mean the exact same thing.

I thought that when there is a subject change (I advise, you conduct), I had to use the subjunctive. Is this an exception?

Exactly. The general rule is that a change in subject triggers the subjunctive. However, verbs of influence are a special exception. Because the person being influenced is already stated as the indirect object (te), Spanish allows you to skip the subjunctive and just use the infinitive.

If I rewrite this sentence using the subjunctive, does the pronoun te move?

No, the pronoun stays exactly where it is. It functions as the indirect object of the main verb (aconsejar). If you switch to the subjunctive structure, it simply becomes Te aconsejo que dirijas la orquesta hoy.

I usually see dirigir translated as to direct. Does it specifically mean to conduct when talking about music?

Yes, in a musical context, dirigir is the standard verb for conducting an orchestra or a choir. Outside of music, it is very versatile and means to direct, to manage, or to run (such as a business, a film, or a project).